Search results for "vector field tomography"

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X-ray Tomography of One-forms with Partial Data

2021

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.

Mathematics - Differential Geometry46F12 44A12 58A10Open set01 natural sciencesinversio-ongelmatintegraaliyhtälötSet (abstract data type)vector field tomographytomografiaFOS: MathematicsNormal operator0101 mathematicsMathematicsx-ray tomographyinverse problemsEuclidean spaceApplied MathematicsMathematical analysisInverse problemunique continuationnormal operatorFunctional Analysis (math.FA)Mathematics - Functional Analysis010101 applied mathematicsComputational MathematicsDifferential Geometry (math.DG)röntgenkuvausTomographyfunktionaalianalyysiAnalysisSIAM Journal on Mathematical Analysis
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Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography

2022

AbstractWe prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.

osittaisdifferentiaaliyhtälötvector field tomographyinverse problemstomografiaApplied MathematicsGeneral MathematicsX-ray tomographyregion of interest tomographyunique continuationinversio-ongelmatAnalysisJournal of Fourier Analysis and Applications
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